Given $ \overrightarrow{OA}\perp\overrightarrow{OC}$, $ m \angle BOC = 3x + 42$, and $ m \angle AOB = 3x + 24$, find $m\angle BOC$. $O$ $A$ $C$ $B$
From the diagram, we see that together ${\angle AOB}$ and ${\angle BOC}$ form ${\angle AOC}$ , so $ {m\angle AOB} + {m\angle BOC} = {m\angle AOC}$ Since we are given that $\overrightarrow{OA}\perp\overrightarrow{OC}$ , we know ${m\angle AOC = 90}$ Substitute in the expressions that were given for each measure: $ {3x + 24} + {3x + 42} = {90}$ Combine like terms: $ 6x + 66 = 90$ Subtract $66$ from both sides: $ 6x = 24$ Divide both sides by $6$ to find $x$ $ x = 4$ Substitute $4$ for $x$ in the expression that was given for $m\angle BOC$ $ m\angle BOC = 3({4}) + 42$ Simplify: $ {m\angle BOC = 12 + 42}$ So ${m\angle BOC = 54}$.